Digital Design Debdeep Mukhopadhyay Associate Professor Dept of

Digital Design Debdeep Mukhopadhyay Associate Professor Dept of Computer Science and Engineering NYU Shanghai and IIT Kharagpur 1

Introduction to Synchronous Sequential Circuits and Iterative Networks 2

Sequential Circuits and Finite-state Machines Sequential circuit: its outputs a function of external inputs as well as stored information Finite-state machine (FSM): abstract model to describe the synchronous sequential machine and its spatial counterpart, the iterative network Serial binary adder example: block diagram, addition process, state table and state diagram Let A denote the state of the adder at ti if the carry 0 is generated at ti-1 Let B denote the state of the adder at ti if the carry 1 is generated at ti-1 3

State Assignment Device with two states capable of storing information regarding the presence or absence of carry: delay element with input Y and output y • State of the device is denote by the state of the output • Two states: y = 0 and y = 1 • The capability of the device to store information is the result of the fact that it takes some time for the input signal Y to pass to the output y – Compare with a combinational gate where the output changes almost immediately. In this device there is some well-defined time required before the input Y passes to the output y. During that time-window, which is usually a clock cycle, the old value stays! • Since the present input value Y of the delay element is equal to its next output value: the input value is referred to as the next state of the delay – y(t+1)=Y(t) Example: assign state y = 0 to state A of the adder and y = 1 to B • The value of y at ti corresponds to the value of the carry generated at ti-1 • Process of assigning the states of a physical device to the states of the serial adder: called state assignment Y = x 1 x 2 + x 1 y + x 2 y • Output value y: referred to as the state variable • Transition/output table for the serial adder: z=x x y 1 2 4

FSM: Definitions FSMs: whose past histories can affect their future behavior in only a finite number of ways • Serial adder: its response to the signals at time t is only a function of these signals and the value of the carry at t-1 – Thus, its input histories can be grouped into just two classes: those resulting in a 1 carry and those resulting in a 0 carry at t • Thus, every finite-state machine contains a finite number of memory devices: which store the information regarding the past input history 5

Synchronous Sequential Machines Input variables: {x 1, x 2, . . , xl} Input configuration, symbol, pattern or vector: ordered l-tuple of 0’s and 1’s Input alphabet: set of p = 2 l distinct input patterns • Thus, input alphabet I = {I 1, I 2, . . , Ip} • Example: for two variables x 1 and x 2 – I = {00, 01, 10, 11} Output variables: {z 1, z 2, . . , zm} Output configuration, symbol, pattern or vector: ordered m-tuple of 0’s and 1’s Output alphabet: set of q = 2 m distinct output patterns • Thus, output alphabet O = {O 1, O 2, . . , Oq} 6

Synchronous Sequential Machines (Contd. ) Set of state variables: {y 1, y 2, . . , yk} Present state: combination of values at the outputs of k memory elements Set S of n = 2 k k-tuples: entire set of states S = {S 1, S 2, . . , Sn} Next state: values of Y’s Synchronization achieved by means of clock pulses feeding the memory devices Initial state: state of the machine before the application of an input sequence to it Final state: state of the machine after the application of the input sequence 7

Memory Elements and Their Excitation Functions To generate the Y’s: memory devices must be supplied with appropriate input values • Excitation functions: switching functions that describe the impact of xi’s and yj’s on the memory-element input • Excitation table: its entries are the values of the memory-element inputs Most widely used memory elements: flip-flops, which are made of latches • Latch: remains in one state indefinitely until an input signals directs it to do otherwise Set-reset of SR latch: 8

SR Latch (Contd. ) Excitation characteristics and requirements: Clocked SR latch: all state changes synchronized to clock pulses • Restrictions placed on the length and frequency of clock pulses: so that the circuit changes state no more than once for each clock pulse 9

Timing the SR latch Tperiod ton • A clock is a periodic signal that is used to keep time in sequential circuits. • Duty Cycle is the ration of ton/Tperiod • We want to keep ton small so that in the same clock pulse only a single computation is performed. • We want to keep Tperiod sufficient so that there is enough time for the next input to be computed. 10

Cascading Latches L 1 L 2 clk When the clock is high both level-sensitive latches are enabled. Consider L 2: If duty-cycle is larger than the time required for the input to L 1 to propagate to the output of L 1, then the input of L 2 gets changed! 11

Trigger or T Latch Value 1 applied to its input triggers the latch to change state Excitations requirements: y(t+1) = Ty’(t) + T’y(t) = T y(t) 12

The JK Latch Unlike the SR latch, J = K = 1 is permitted: when it occurs, the latch acts like a trigger and switches to the complement state Excitation requirements: 13

The D Latch The next state of the D latch is equal to its present excitation: y(t+1) = D(t) 14

Clock Timing Clocked latch: changes state only in synchronization with the clock pulse and no more than once during each occurrence of the clock pulse Duration of clock pulse: determined by circuit delays and signal propagation time through the latches • Must be long enough to allow latch to change state, and • Short enough so that the latch will not change state twice due to the same excitation Excitation of a JK latch within a sequential circuit: • Length of the clock pulse must allow the latch to generate the y’s • But should not be present when the values of the y’s have propagated through the combinational circuit 15

JK-Latch in Real Life in Toggle Mode 1 1 1 0 1 16

JK-Latch in Real Life in Toggle Mode 1 1 1 0 1 17

JK-Latch in Real Life in Toggle Mode 1 1 1 0 One needs to disable the clock! Else… 18

JK-Latch in Real Life in Toggle Mode 1 1 1 0 1 1 1 The transition of is for Qbar is for a small time interval. 19

Master-slave Flip-flop Master-slave flip-flop: a type of synchronous memory element that eliminates the timing problems by isolating its inputs from its outputs Master-slave SR flip-flop: Master-slave JK flip-flop: since master-slave SR flip-flop suffers from the problem that both its inputs cannot be 1, it can be converted to a JK flip-flip 20

Master-slave JK Flip-flop with Additional Inputs Direct set and clear inputs: override regular input signals and clock • • • To set the slave output to 0: make set = 1 and clear = 0 To set the slave output to 1: make set = 0 and clear = 1 Assigning 0 to both set and clear: not allowed Assigning 1 to both set and clear: normal operation Useful in design of counters and shift registers 21

1’s Catching and 0’s Catching SR and JK flip-flops suffer from 1’s catching and 0’s catching Master latch is transparent when the clock is high • Consider the latch holding its value, ie. J=0, K=0. • When the output of the slave latch is at 0 and the J input has a static-0 hazard (a transient glitch to 1) after the clock has gone high: then the master latch catches this set condition – It then passes the 1 to the slave latch when the clock goes low • Similarly, when the output of the slave latch is at 1 and the K input has a static-0 hazard after the clock has gone high: then the master latch catches this reset condition – It then passes the 0 to the slave latch when the clock goes low 22

D flip-flop Master-slave D flip-flop avoids the above problem: even when a static hazard occurs at the D input when the clock is high, the output of the master latch reverts to its old value when the glitch goes away 23

Edge-triggered Flip-flop Positive (negative) edge-triggered D flip-flip: stores the value at the D input when the clock makes a 0 -> 1 (1 -> 0) transition • Any change at the D input after the clock has made a transition does not have any effect on the value stored in the flip-flop A negative edge-triggered D flip-flop: • When the clock is high, the output of the bottommost (topmost) NOR gate is at D’ (D), whereas the S-R inputs of the output latch are at 0, causing it to hold previous value • When the clock goes low, the value from the bottommost (topmost) NOR gate gets transferred as D (D’) to the S (R) input of the output latch – Thus, output latch stores the value of D • If there is a change in the value of the D input after the clock has made its transition, the bottommost NOR gate attains value 0 – However, this cannot change the SR inputs of the output latch 24

Edge-triggered Flip-flop D’ D Changes to the input D does not affect the output y and y’ when the clock is high. 0 1 0 D’ 25

Edge-triggered Flip-flop D’ D D’ Output RS-Latch stores a 0 or 1, depending on the value of D. 0 D What happens when D changes and clock is low? D’ 26

Synthesis of Synchronous Sequential Circuits Main steps: 1. From a word description of the problem, form a state diagram or table 2. Check the table to determine if it contains any redundant states • If so, remove them (Chapter 10) 3. Select a state assignment and determine the type of memory elements 4. Derive transition and output tables 5. Derive an excitation table and obtain excitation and output functions from their respective tables 6. Draw a circuit diagram 27

Sequence Detector One-input/one-output sequence detector: produces output value 1 every time sequence 0101 is detected, else 0 • Example: 010101 -> 000101 State diagram and state table: Transition and output tables: 28

Sequence Detector (Contd. ) Excitation and output maps: z = xy 1 y 2’ y 1 = x’y 1 y 2 + xy 1’y 2 + xy 1 y 2’ y 2 = y 1 y 2’ + x’y 1’ + y 1’y 2 Logic diagram: 29

Sequence Detector (Contd. ) Another state assignment: z = xy 1 y 2 Y 1 = x’y 1 y 2’ + xy 2 Y 2 = x’ 30

Binary Counter One-input/one-output modulo-8 binary counter: produces output value 1 for every eighth input 1 value State diagram and state table: 31

Binary Counter (Contd. ) Transition and output tables: Excitation table for T flip-flops and logic diagram: T 1 = x T 2 = xy 1 T 3 = xy 1 y 2 z = xy 1 y 2 y 3 32

Implementing the Counter with SR Flipflops Transition and output tables: Excitation table for SR flip-flops and logic diagram: • Trivially extensible to modulo-16 counter S 1 = xy 1’ R 1 = xy 1 S 2 = xy 1 y 2’ R 2 = xy 1 y 2 S 3 = xy 1 y 2 y 3’ R 3 = z = xy 1 y 2 y 3 33

Parity-bit Generator Serial parity-bit generator: receives coded messages and adds a parity bit to every m-bit message • Assume m = 3 and even parity State diagram and state table: J 1 = y 2 K 1 = y 2’ J 2 = y 1’ K 2 = y 1 J 3 = xy 1’ + xy 2 K 3 = x + y 2’ z = y 2’y 3 34

Sequential Circuit as a Control Element Control element: streamlines computation by providing appropriate control signals Example: digital system that computes the value of (4 a + b) modulo 16 • • • a, b: four-bit binary number X: register containing four flip-flops x: number stored in X Register can be loaded with: either b or a + x Addition performed by: a four-bit parallel adder K: modulo-4 binary counter, whose output L equals 1 whenever the count is 3 modulo 4 35

Example (Contd. ) Sequential circuit M: • • Input u: initiates computation Input L: gives the count of K Outputs: , , , z When = 1: contents of b transferred to X When = 1: values of x and a added and transferred back to X When = 1: count of K increased by 1 z = 1: whenever final result available in X 36

Example (Contd. ) Sequential circuit M: • K, u, z: initially at 0 • When u = 1: computation starts by setting = 1 – Causes b to be loaded into X • To add a to x: set = 1 and = 1 to keep track of the number of times a has been added to x • After four such additions: z = 1 and the computation is complete • At this point: K = 0 to be ready for the next computation State diagram: 37

Example (Contd. ) State assignment, transition table, maps and logic diagram: = y 1’y 2 = = y 1 y 2 z = y 1 y 2’ Y 1 = y 2 Y 2 = y 1’y 2 + uy 1’ + L’y 2 38